Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{n^2 - 2n - 8}{n^2 - 4n - 12}$
First factor the expressions in the numerator and denominator. $ \dfrac{n^2 - 2n - 8}{n^2 - 4n - 12} = \dfrac{(n - 4)(n + 2)}{(n - 6)(n + 2)} $ Notice that the term $(n + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(n + 2)$ gives: $x = \dfrac{n - 4}{n - 6}$ Since we divided by $(n + 2)$, $n \neq -2$. $x = \dfrac{n - 4}{n - 6}; \space n \neq -2$